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Theorem srglz 18350
 Description: The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
Hypotheses
Ref Expression
srgz.b 𝐵 = (Base‘𝑅)
srgz.t · = (.r𝑅)
srgz.z 0 = (0g𝑅)
Assertion
Ref Expression
srglz ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )

Proof of Theorem srglz
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgz.b . . . . . . . 8 𝐵 = (Base‘𝑅)
2 eqid 2610 . . . . . . . 8 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2610 . . . . . . . 8 (+g𝑅) = (+g𝑅)
4 srgz.t . . . . . . . 8 · = (.r𝑅)
5 srgz.z . . . . . . . 8 0 = (0g𝑅)
61, 2, 3, 4, 5issrg 18330 . . . . . . 7 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ (mulGrp‘𝑅) ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦(+g𝑅)𝑧)) = ((𝑥 · 𝑦)(+g𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
76simp3bi 1071 . . . . . 6 (𝑅 ∈ SRing → ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦(+g𝑅)𝑧)) = ((𝑥 · 𝑦)(+g𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
87r19.21bi 2916 . . . . 5 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦(+g𝑅)𝑧)) = ((𝑥 · 𝑦)(+g𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
98simprd 478 . . . 4 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))
109simpld 474 . . 3 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → ( 0 · 𝑥) = 0 )
1110ralrimiva 2949 . 2 (𝑅 ∈ SRing → ∀𝑥𝐵 ( 0 · 𝑥) = 0 )
12 oveq2 6557 . . . 4 (𝑥 = 𝑋 → ( 0 · 𝑥) = ( 0 · 𝑋))
1312eqeq1d 2612 . . 3 (𝑥 = 𝑋 → (( 0 · 𝑥) = 0 ↔ ( 0 · 𝑋) = 0 ))
1413rspcv 3278 . 2 (𝑋𝐵 → (∀𝑥𝐵 ( 0 · 𝑥) = 0 → ( 0 · 𝑋) = 0 ))
1511, 14mpan9 485 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Mndcmnd 17117  CMndccmn 18016  mulGrpcmgp 18312  SRingcsrg 18328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-srg 18329 This theorem is referenced by:  srgmulgass  18354  srgrmhm  18359
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